A099023 -id:A099023 - cp's OEIS frontend

A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

1, 1, 4, 46, 1024, 36976, 1965664, 144361456, 13997185024, 1731678144256, 266182076161024, 49763143319190016, 11118629668610842624, 2925890822304510631936, 895658946905031792553984, 315558279782214450517374976, 126780706777739389745128013824

Offset: 0

N. J. A. Sloane, Don Knuth

Also central terms of the triangle in A008280. - Reinhard Zumkeller, Nov 01 2013

Conjecture: taking the sequence modulo an integer k gives an eventually purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 9 begins [1, 1, 4, 1, 7, 4, 1, 7, 4, 1, 7, ...] with an apparent period [4, 1, 7] of length 3 = phi(9)/2 beginning at a(2). - Peter Bala, May 08 2023

V. I. Arnold, Springer numbers and Morsification spaces. J. Algebraic Geom. 1 (1992), no. 2, 197-214.
L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A084938, A002832. For a signed version see A099023.
Related polynomials in A098277.
A diagonal of A323834.
Cf. A005799.

a000657 n = a008280 (2 * n) n  -- Reinhard Zumkeller, Nov 01 2013

Digits := 40: rr := array(1..40,1..40): rr[1,1] := 1: for i from 1 to 39 do rr[i+1,1] := subs(x=0,diff(1+tan(x),x$i)): od: for i from 2 to 40 do for j from 2 to i do rr[i,j] := rr[i,j-1]-(-1)^i*rr[i-1,j-1]: od: od: [seq(rr[2*i-1,i],i=1..20)];
# Alternatively after Alois P. Heinz in A000111:
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> b(n, n): seq(a(n), n = 0..15); # Peter Luschny, Oct 27 2017

max = 20; rr[1, 1] = 1; For[i = 1, i <= 2*max - 1, i++, rr[i + 1, 1] = D[1 + Tan[x], {x, i}] /. x -> 0]; For[i = 2, i <= 2*max, i++, For[j = 2, j <= i, j++, rr[i, j] = rr[i, j - 1] - (-1)^i*rr[i - 1, j - 1]]]; Table[rr[2*i - 1, i], {i, 1, max}] (* Jean-François Alcover, Jul 10 2012, after Maple *)
T[n_,0] := KroneckerDelta[n,0]; T[n_,k_] := T[n,k]=T[n,k-1]+T[n-1,n-k]; Table[T[2n,n], {n,0,16}] (* Oliver Seipel, Nov 24 2024, after Peter Luschny *)

a(n):=(-1)^(n)*sum(binomial(n,k)*euler(n+k),k,0,n); /* Vladimir Kruchinin, Apr 06 2015 */

# Algorithm of L. Seidel (1877)
def A000657_list(n) :
    R = []; A = {-1:0, 0:1}
    k = 0; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) :
            Am += A[k]; A[k] = Am; k += e
        if e < 0 :
            R.append(A[0])
    return R
A000657_list(30)  # Peter Luschny, Apr 02 2012

Row sums of triangle, read by rows, [0, 1, 4, 9, 16, 25, 36, 49, ...] DELTA [1, 2, 6, 5, 11, 8, 16, 11, 21, 14, ...] where DELTA is Deléham's operator defined in A084938.
G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*1x/(1-1*3x/(1-2*5x/(1-2*7x/(1-3*9x/...))))). - Ralf Stephan, Sep 09 2004
G.f.: 1/G(0) where G(k) = 1 - x*(8*k^2+4*k+1) - x^2*(k+1)^2*(4*k+1)*(4*k+3)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: G(0)/(1-x), where G(k) = 1 - x^2*(k+1)^2*(4*k+1)*(4*k+3)/( x^2*(k+1)^2*(4*k+1)*(4*k+3) - (1 - x*(8*k^2+4*k+1))*(1 - x*(8*k^2+20*k+13))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 01 2014
a(n) = (-1)^(n)*Sum_{k=0..n} C(n,k)*Euler(n+k). - Vladimir Kruchinin, Apr 06 2015
a(n) ~ 2^(4*n+5/2) * n^(2*n+1/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Apr 06 2015
Conjectural e.g.f. as a continued fraction: 1/(1 - (1 - exp(-2*t))/(2 - (1 - exp(-4*t))/(1 - (1 - exp(-6*t))/(2 - (1 - exp(-8*t))/(1 - ... )))) = 1 + t + 4*t^2/2! + 46*t^3/3! + .... Cf. A005799. - Peter Bala, Dec 26 2019

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001

Corrected by Sean A. Irvine, Dec 22 2010

A163982 Real part of the coefficient [x^n] of the expansion of (1+i)/(1-iexp(x)) - 1 multiplied by 2n!, where i is the imaginary unit.

Original entry on oeis.org

-2, -1, 1, 2, -5, -16, 61, 272, -1385, -7936, 50521, 353792, -2702765, -22368256, 199360981, 1903757312, -19391512145, -209865342976, 2404879675441, 29088885112832, -370371188237525, -4951498053124096, 69348874393137901

Offset: 0

Roger L. Bagula, Aug 07 2009

sign

The sequence is a signed variant of A163747 and starts with a two instead of a zero.
From Paul Curtz, Mar 20 2013: (Start)
-a(n) and successive differences are:
    2,    1,    -1,     -2,      5,     16,    -61, -272;
   -1,   -2,    -1,      7,     11,    -77,   -211, 1657, ...
   -1,    1,     8,      4,    -88,   -134,   1868, 4894, ...
    2,    7,    -4,    -92,    -46,    -46,   2002, 3026, ...
    5,  -11,   -88,     46,   2048,   1024, -72928, ...
  -16,  -77,   134,   2002,  -1024, -73952, -36976, ...
  -61,  211,  1868,  -3026, -72928, ...
  272, 1657, -4894, -69902, ...
This is an autosequence: The inverse binomial transform (left column of the array of differences) is the signed sequence. The main diagonal 2, -2, 8, -92, ... doubles the entries of the first upper diagonal 1, -1, 4, -46, ... = A099023(n).
Sum of the antidiagonals: 2, 0, -4, 0, 32, ... = 2*A155585(n+1). (End)

G. C. Greubel, Table of n, a(n) for n = 0..480
Toufik Mansour, Howard Skogman, Rebecca Smith, Passing through a stack k times with reversals, arXiv:1808.04199 [math.CO], 2018.

Cf. A163747.

A163982 := n -> -2^n*(euler(n,1/2)+euler(n,1)): # Peter Luschny, Nov 25 2010
A163982 := proc(n)
    (1+I)/(1-I*exp(x))-1 ;
    coeftayl(%,x=0,n) ;
    Re(%*2*n!) ;
end proc; # R. J. Mathar, Mar 26 2013

f[t_] = (1 + I)/(1 - I*Exp[t]) - 1; Table[Re[2*n!*SeriesCoefficient[Series[f[t], {t, 0, 30}], n]], {n, 0, 30}]
max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 + x + (4*k+3)*(k+1)*x^2 /( 1 + (4*k+5)*(k+1)*x^2 / g[k+1]); gf = -2 - x/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jan 22 2015, after Sergei N. Gladkovskii *)
With[{nn = 50}, CoefficientList[Series[(-2)*Exp[t/2]*Cosh[t/2]/Cosh[t], {t, 0, nn}], t]*Range[0, nn]!] (* G. C. Greubel, Aug 24 2017 *)

t='t+O('t^10); Vec(serlaplace((-2)*exp(x/2)*cosh(x/2)/cosh(x))) \\ G. C. Greubel, Aug 24 2017

Let ((1 + i)/(1 - i*exp(t)) - 1) = a(n) + I*b(n); abs(a(n)) = abs(b(n)).
a(n) = -2^n*(E_{n}(1/2) + E_{n}(1)), E_{n}(x) Euler polynomial. - Peter Luschny, Nov 25 2010
E.g.f.: -(1/cosh(x) + tanh(x)) - 1. - Sergei N. Gladkovskii, Dec 11 2013
G.f.: -2 - x/W(0), where W(k) = 1 + x + (4*k+3)*(k+1)*x^2 /( 1 + (4*k+5)*(k+1)*x^2 /W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 22 2015
E.g.f.: (-2)*exp(x/2)*cosh(x/2)/cosh(x). - G. C. Greubel, Aug 24 2017

A009744 Expansion of e.g.f. tan(x)*sin(x) (even powers only).

Original entry on oeis.org

0, 2, 4, 62, 1384, 50522, 2702764, 199360982, 19391512144, 2404879675442, 370371188237524, 69348874393137902, 15514534163557086904, 4087072509293123892362, 1252259641403629865468284, 441543893249023104553682822, 177519391579539289436664789664

Offset: 0

R. H. Hardin

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A029582, A099023.

seq((2*i)!*coeff(series(tan(x)*sin(x),x,30),x,2*i),i=0..14); # Peter Luschny, Jul 14 2012

nn = 30; t = Range[0, nn]! CoefficientList[Series[Tan[x]*Sin[x], {x, 0, nn}], x]; Take[t, {1, nn, 2}] (* T. D. Noe, Jul 15 2012 *)

x='x+O('x^50); v=Vec(serlaplace(tan(x)*sin(x))); concat([0], vector(#v\2,n,v[2*n-1])) \\ G. C. Greubel, Mar 04 2018

# Variant of an algorithm of L. Seidel (1877) with a(0) = 1.
def A009744_list(n) :
    dim = 2*n; E = matrix(ZZ, dim); E[0, 0] = 1
    for m in (1..dim-1) :
        if m % 2 == 0 :
            E[m, 0] = 1;
            for k in range(m-1, -1, -1) :
                E[k, m-k] = E[k+1, m-k-1] - E[k, m-k-1]
        else :
            E[0, m] = 1;
            for k in range(1, m+1, 1) :
                E[k, m-k] = E[k-1, m-k+1] + E[k-1, m-k]
    return [(-1)^(k//2)*E[0,k] for k in range(dim) if is_even(k)]
A009744_list(14)  # Peter Luschny, Jul 14 2012

G.f.: 1/G(0) - 1/(1+x) where G(k) = 1 - x*(2*k+1)^2/(1 - x*(2*k+2)^2/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: 1/G(0) - 1/(1+x) where G(k) = 1 - x*(k+1)^2/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 09 2013
a(n) ~ (2*n)! * 4^(n+1) / Pi^(2*n+1). - Vaclav Kotesovec, Jan 24 2015
Conjectural o.g.f.: Sum_{n >= 0} 4*x/2^n * Sum_{k = 0..n} (-1)^k*(k+1)*binomial(n,k)/( (1 + x*(2*k + 1)^2)*(1 + x*(2*k + 3)^2) ) = 2*x + 4*x^2 + 62*x^3 + 1384*x^4 + .... - Peter Bala, Mar 03 2015
From Peter Luschny, Jun 13 2021: (Start)
a(n) = (-1)^n*(Euler(2*n) - 1).
a(n) ~ 4^(2*n + 3/2)*exp(1/(24*n) - 2*n)*(n/Pi)^(2*n + 1/2). (End)

Extended and signs tested by Olivier Gérard, Mar 15 1997

A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows.

Original entry on oeis.org

1, 0, 1, -1, -1, 0, 0, -1, -2, -2, 5, 5, 4, 2, 0, 0, 5, 10, 14, 16, 16, -61, -61, -56, -46, -32, -16, 0, 0, -61, -122, -178, -224, -256, -272, -272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0, 0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936

Offset: 0

Paul Curtz, Mar 08 2014

			The triangle T(n,k) begins:
                      1
                    0   1
                 -1  -1   0
                0  -1  -2  -2
              5   5   4   2   0
             ...
The array read as a table, A(n,k) = T(n+k, k), starts:
     1,    1,    0,   -2,    0,   16,    0, -272,    0, ...
     0,   -1,   -2,    2,   16,  -16, -272,  272, ...
    -1,   -1,    4,   14,  -32, -256,  544, ...
     0,    5,   10,  -46, -224,  800, ...
     5,    5,  -56, -178, 1024, ...
     0,  -61, -122, 1202, ...
   -61,  -61, 1324, ...
     0, 1385, ...
  1385, ...
  ...
For the above table, we have A(n,k) = (-1)^(n+k)*A236935(n,k) for n, k >= 0. It has joint e.g.f. 2*exp(-x)/(1 + exp(-2*(x+y))). - _Petros Hadjicostas_, Feb 21 2021

L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187; see Beilage 4 (p. 187).

Unsigned version is A008280.

Cf. A008281, A099023, A108040, A122045, A155585, A236935.

t[0, 0] = 1; t[n_, m_] /; nJean-François Alcover, Dec 30 2014 *)

T(n,m):=sum(binomial(m,k)*euler(n-m+k),k,0,m); /* Vladimir Kruchinin, Apr 06 2015 */

a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1) /* A122045 */
T(n, k) = (-1)^n*sum(i=0, k, (-1)^i*binomial(k, i)*a(n-i)) /* Petros Hadjicostas, Feb 21 2021 */
/* Second PARI program (same a(n) for A122045 as above) */
T(n, k) = sum(i=0, k, binomial(k, i)*a(n-k+i)) /* Petros Hadjicostas, Feb 21 2021 */

a(n) = A057077(n)*A008280(n) by rows.
a(n) is the increasing antidiagonals of the difference table of A155585(n).
Central column of triangle: A099023(n).
Right main diagonal of triangle: A155585(n) (see A009006(n)).
Left main diagonal of triangle: A122045(n).
T(n,m) = Sum_{k=0..n} binomial(m,k)*Euler(n-m+k) for 0 <= m <= n. - Vladimir Kruchinin, Apr 06 2015 [The summation only needs to go from k=0 to k=m because of binomial(m,k).]
T(n,k) = (-1)^n*A236935(n-k,k) for 0 <= k <= n, where the latter is read as a square array. - Petros Hadjicostas, Feb 21 2021

A029582 E.g.f. sin(x) + cos(x) + tan(x).

Original entry on oeis.org

1, 2, -1, 1, 1, 17, -1, 271, 1, 7937, -1, 353791, 1, 22368257, -1, 1903757311, 1, 209865342977, -1, 29088885112831, 1, 4951498053124097, -1, 1015423886506852351, 1, 246921480190207983617, -1, 70251601603943959887871, 1, 23119184187809597841473537

Offset: 0

N. J. A. Sloane

sign

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A009744, A099023.

nn = 30; Range[0, nn]! CoefficientList[Series[Tan[x] + Sin[x] + Cos[x], {x, 0, nn}], x] (* T. D. Noe, Jul 16 2012 *)

# Variant of an algorithm of L. Seidel (1877).
def A029582_list(n) :
    dim = 2*n; E = matrix(ZZ, dim); E[0, 0] = 1
    for m in (1..dim-1) :
        if m % 2 == 0 :
            E[m, 0] = 1;
            for k in range(m-1, -1, -1) :
                E[k, m-k] = E[k+1, m-k-1] - E[k, m-k-1]
        else :
            E[0, m] = 1;
            for k in range(1, m+1, 1) :
                E[k, m-k] = E[k-1, m-k+1] + E[k-1, m-k]
    return [(-1)^(k//2)*E[k,0] for k in range(dim)]
A029582_list(15)  # Peter Luschny, Jul 14 2012

G.f.: (1+x)/(1+x^2)+x/T(0) where T(k)= 1 - (k+1)*(k+2)*x^2/T(k+1) ; (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 12 2013

G.f.: (1+x)/(1+x^2)+x/G(0) where G(k) = 1 - 2*x^2*(4*k^2+4*k+1) - 4*x^4*(k+1)^2*(4*k^2+8*k+3)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 12 2013

A236935 The infinite Seidel matrix H read by antidiagonals upwards; H = (H(n,k): n,k >= 0).

Original entry on oeis.org

1, 0, -1, -1, -1, 0, 0, 1, 2, 2, 5, 5, 4, 2, 0, 0, -5, -10, -14, -16, -16, -61, -61, -56, -46, -32, -16, 0, 0, 61, 122, 178, 224, 256, 272, 272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0, 0, -1385, -2770, -4094, -5296, -6320, -7120, -7664, -7936, -7936, -50521, -50521, -49136, -46366, -42272, -36976, -30656, -23536, -15872, -7936, 0

Offset: 0

N. J. A. Sloane, Feb 17 2014

This is, in essence, a signed version of the triangle in A008280.

			Array begins:
     1   -1    0    2    0 -16   0 272 0 ...
     0   -1    2    2  -16 -16 272 272 ...
    -1    1    4  -14  -32 256 544 ...
     0    5  -10  -46  224 800 ...
     5   -5  -56  178 1024 ...
     0  -61  122 1202 ...
   -61   61 1324 ...
     0 1385 ...
  1385 ...
  ...

D. Dumont and G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Preprint, Annotated scanned copy.
D. Dumont and G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Annals of Discrete Mathematics, 6 (1980), 77-87.
Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.
Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, European Journal of Combinatorics, 42 (2014), 243-260.
L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187; see Beilage 4 (p. 187).

Cf. A008280, A099023, A122045, A155585, A239005.

a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1) /* A122045 */
H(n,k) = sum(i=0, k, (-1)^i*binomial(k,i)*a(n+k-i)) /* Petros Hadjicostas, Feb 21 2021 */
/* Second PARI program (same a(n) for A122045 as above) */
H(n,k) = (-1)^(n+k)*sum(i=0, k, binomial(k,i)*a(n+i)) /* Petros Hadjicostas, Feb 21 2021 */

From Petros Hadjicostas, Feb 20 2021: (Start)
H(n,0) = A122045(n).
H(0,k) = (-1)^n*A155585(n).
H(n,k) = Sum_{i=0..n} binomial(n,i)*H(0,k+i).
H(n,k) = Sum_{i=0..k} (-1)^i*binomial(k,i)*H(n+k-i,0).
H(n,n) = A099023(n).
Bivariate e.g.f.: Sum_{n,k>=0} H(n,k)*(x^n/n!)*(y^k/k!) = 2*exp(x)/(1 + exp(2*(x+y))).
H(n,k) = (-1)^(n+k)*A239005(n+k,k), where the latter is a triangle.
H(n,k) = -A008280(n+k,k) if ((n+k) mod 4) == 1 or 2, and H(n,k) = A008280(n+k,k) if ((n+k) mod 4) == 3 or 0, provided A008280 is read as a triangle. (End)

More terms from Petros Hadjicostas, Feb 21 2021

A213736 Triangle read by rows, coefficients of the Swiss-Knife median polynomials M_{n}(x) in descending order of powers.

Original entry on oeis.org

1, 1, -1, -1, 1, -2, -5, 6, 4, 1, -3, -12, 29, 57, -72, -46, 1, -4, -22, 80, 261, -660, -1264, 1608, 1024, 1, -5, -35, 170, 775, -2941, -9385, 23880, 45620, -58080, -36976, 1, -6, -51, 310, 1815, -9186, -41033, 156618, 498660, -1269720, -2425056, 3087648

Offset: 0

Peter Luschny, Jun 19 2012

M(n,0) = M(n,1) = A099023(n) = (-1)^n*A000657(n).

			M(0,x) = 1,
M(1,x) = x^2-x-1,
M(2,x) = x^4-2*x^3-5*x^2+6*x+4,
M(3,x) = x^6-3*x^5-12*x^4+29*x^3+57*x^2-72*x-46.

Peter Luschny, An old operation on sequences: the Seidel transform.

Cf. A153641, A162660.

A213736_triangle := proc(n) local A, len, k, m, sk_poly;
len := 2*n-1; A := array(0..len,0..len);
sk_poly := proc(n, x) local v, k;
add(`if`((k+1)mod 4 = 0,0,(-1)^iquo(k+1,4))*2^iquo(-k,2)*
add((-1)^v*binomial(k,v)*(v+x+1)^n,v=0..k),k=0..n) end:
for m from 0 to len do A[m,0] := sk_poly(m,x);
   for k from m-1 by -1 to 0 do
       A[k,m-k] := A[k+1,m-k-1] - A[k,m-k-1] od od;
seq(print(seq(coeff(A[k,k],x,2*k-i),i=0..2*k)),k=0..n-1) end:
A213736_triangle(5);

cp's OEIS Frontend

A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

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A163982 Real part of the coefficient [x^n] of the expansion of (1+i)/(1-i*exp(x)) - 1 multiplied by 2*n!, where i is the imaginary unit.

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A009744 Expansion of e.g.f. tan(x)*sin(x) (even powers only).

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Maple

Mathematica

PARI

Sage

Formula

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A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows.

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Mathematica

Maxima

PARI

Formula

A029582 E.g.f. sin(x) + cos(x) + tan(x).

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Mathematica

Sage

Formula

A236935 The infinite Seidel matrix H read by antidiagonals upwards; H = (H(n,k): n,k >= 0).

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PARI

Formula

Extensions

A213736 Triangle read by rows, coefficients of the Swiss-Knife median polynomials M_{n}(x) in descending order of powers.

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A163982 Real part of the coefficient [x^n] of the expansion of (1+i)/(1-iexp(x)) - 1 multiplied by 2n!, where i is the imaginary unit.